PH3203 - Preliminary Lecture 06

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Издатель
In this lecture I describe the solution for the radial equation for an isotropic harmonic oscillator in three dimensions. This can be easily solved using Cartesian coordinates - which allows us to determine the energy eigenvalues with ease. We then go on to solve the equation in spherical polar coordinates, and investigate how the singularities of the resulting equation leads to prediction about its behavior of the wave function. We show how the energy quantization condition follows from this . Having obtained two distinct expressions for energy using different sets of quantum numbers (in Cartesian and spherical polar coordinates), I then show that they lead to the same energy levels and degeneracies. I finally discuss the connection of the radial equation with an important equation of mathematical physics - the confluent hypergeometric equation. I show the connection of the solutions - the confluent hypergeometric series, of the CHE with the associated Laguerre polynomials and the Herrmite polynomials.
Категория
Новости 3D Печати
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